See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/317623699
Method Of Weighted Residual For Periodic Boundary Value Problem:
Galerkin’s Method Approach.
Article · January 2013
CITATIONS
READS
0
709
4 authors, including:
Nnaemeka Obuka
Pius Chukwukelue Onyechi
Enugu State University of Science and Technology
Nnamdi Azikiwe University, Awka
21 PUBLICATIONS 85 CITATIONS
45 PUBLICATIONS 135 CITATIONS
SEE PROFILE
SEE PROFILE
Ndubuisi C. Okoli
Nnamdi Azikiwe University, Awka
5 PUBLICATIONS 4 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Polymer based graphene composite View project
Life Cycle Cost Analysis of a Diesel/Photovoltaic Hybrid Power Generating System View project
All content following this page was uploaded by Pius Chukwukelue Onyechi on 05 September 2017.
The user has requested enhancement of the downloaded file.
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
Method Of Weighted Residual For Periodic Boundary Value
Problem: Galerkin’s Method Approach
Obuka, Nnaemeka S.P1., Onyechi, Pius C1., Okoli, Ndubuisi C1., and Ikwu, Gracefield.R.O2.
1
Department of Industrial and Production Engineering, Nnamdi Azikiwe University, Awka, Nigeria.
2
Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria.
Corresponding Author’s e-mail: silvermeks7777@yahoo.com
Abstract
Method of weighted residual has been one of the foremost approximation solution to partial differential equation
problems. In this paper a time dependent and boundary-valued strain model obtained from a PPC/CaCO3 composite
experimental data was analyzed using the method of weighted residual. The Galerkin’s approach of this approximate
method was applied to the model at a constant stress of 17.33 MPa. The discretization of the composite material was
therefore, designed to take the form of a linear and one dimensional problem setting.
Keywords: Method of Weighted Residual, PPC/CaCO3 Composite, Galerkin’s Method, Time Dependent,
Boundary-Value.
1. Introduction
Modern structural designs increasingly incorporate man made composite materials in applications that require
components with special material properties which are unavailable from conventional metals or alloys. From the
structural mechanics point of view, composites are typically used to improve either the stiffness-to-weight ratio or
the strength-to-weight ratio of a structural member. The study of composite materials and structures can be
undertaken from two different approaches; micromechanical and macromechanical [1]. The micromechanical
approach which is the main interest of this paper, the properties of the constituent fiber and matrix materials and their
interactions through stress-strain constitutive relations are analyzed in order to predict the overall behavior of the
composite structural member. The composite material under analysis of its stress-strain property is a PP/CaCO3
composite, with the employment of Galerkin’s method of weighted residual of finite element analysis.
Prior to development of the Finite Element Method, there existed an approximation technique for solving differential
equations called the Method of Weighted Residual (MWR) [2]. The history of the criteria to various choices of
weighting functions of the approximate methods could be found in Table 1.
16
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
TABLE I: History of Approximate Methods
Date
Investigator
Method
1915
Galerkin [9]
Galerkin method
1921
Pohlhausen
Integral method
1923
Biezeno and Koch
Subdomain method
1928
Picone
Method of least square
1932
Kravchuk [10]
Method of moments
1933
Kantorovich [11]
Method of reduction to ordinary differential equations
1937
Frazer, Jones and Skan [12]
Collocation method
1938
Poritsky [13]
Method of reduction to ordinary differential equations
1940
Repman [14]
Convergence of Galerkin’s method
1941
Bickley [15]
Collocation, Galerkin, least squares for initial-value problems
1942
Keldysh [16]
Convergence of Galerkin’s method, steady state
1947
Yamada [17]
Method of moments
1949
Faedo
Convergence of Galerkin’s method unsteady state
1953
Green [18]
Convergence of Galerkin’s method unsteady state
1956
Crandall [4]
Unification as method of weighted residual
The method of weighted residuals is an engineer’s tool for finding approximate solutions to the equations of change
of distributed systems. The method is applicable but not limited to nonlinear and nonself adjoint problems, one of its
most attractive features [3]. MWR includes many approximation methods that are being used currently. It provides a
vantage point from which it is easy to see the unity of these methods as well as the relationships between them.
Some of the best early treatments of MWR are those by Crandall [4], who coined the name method of weighted
residuals, Ames [5] and Collatz [6], who called these methods error-distribution principles.
Suppose we have a linear differential operator ‘D’ acting on a function ‘u’ to produce a product ‘p’.
We wish to approximate u by a function ũ, which is a linear combination of basis functions chosen from a linearly
dependent set, that is;
Now, when substituted into the differential operator, D, the result of the operations is not, in general, p(x). Hence an
17
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
error or residual will exist:
The notion in the MWR is to force the residual to zero in some average sense over the domain, thus;
Where the number of weight functions Wi is exactly equal the number of unknown constants ai in ũ. The result is a
set of n algebraic equations for the unknown constants ai. There are (at least) five MWR sub-methods according to
the choices for the Wi’s, these include; (1) Collocation method (2) Sub-domain method (3) Least square method (4)
Method of moments and (5) Galerkin method. In this paper, we are going to apply the Galerkin’s method in
analyzing our obtained boundary valued data. Boundary value problems with periodic conditions appear in a number
of applications, particularly in the homogenization of composite materials with a periodic microstructure [7, 8].
When such problems are soved numerically, the periodicity condition is often imposed strongly, i.e., the solution
values on periodic edges are required to match exactly.
2. Analysis of Data: MWR the Galerkin’s Method
The strain model obtained from a PPC/CaCO3 composite experimental data at a constant stress of 17.33 MPa is
given as follows;
Under the boundary conditions;
Where
ℰ = Strain,
t = Time in hours,
= Stress.
Our initial step is to take the first and the second derivatives of the given model, thus;
18
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
Expressing the last term on the right hand-side of equation (7) as equal to zero at t = 0, hence;
We treat this composite material under constant stress and variable strain as a linear and one dimensional problem, at
fixed boundary conditions, as illustrated in Fig (1) below;
F(x)
ℰ (0,s)
ℰ (t,s)
t=0
x
=
Fig (1)
Discretizing this composite into six equal elements and seven nodes, as in Fig (2) below, hence, at equal time
intervals of 0.5hrs for each element, thus;
1
2
.
[1]
•
3
•
[2]
•
4
[3]
5
•
[4]
6
[5]
7
•
[6]
.
•
Fig (2)
Therefore this system equation (8) can be modeled with a one dimensional form of Poisson’s equation, which is
expressed as,
From equation (8) f(x) = 0.0050 and equation (9) can be expressed as;
19
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
Therefore,
Fig (3) shows an individual element and Fig (4) shows the approximation function (element equation) used in
expressing the strain distribution along the element, and it’s related as;
Node 1
Node 2
•
•
Fig (3)
ℰ
ℰ̅
ℰ
Fig (4)
Where N1 and N2 are linear interpolation functions expressed as, thus;
Equation (12) can be substituted into equation (11) but, as equation (11) is not the exact solution, the hand-side of the
resulting equation will not be equal to zero, but will be equal to a residual “R”, thus;
The method of weighted residual (MWR) consists of finding a minimum for the residual “R” according to the
general formula, below;
20
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
Where
www.iiste.org
D = the solution domain
Wi = linear independent weighting functions
For the
the interpolation functions
can be employed to substitute it as the weighting functions. When these
are substituted into equation (16), the result is referred to as the Galerkin’s approach, thus;
Therefore,
Re-expressed as;
Therefore,
Applying integration by parts, first on the left hand-side of the above equation (20), hence making our choices of
terms;
Therefore,
The first term on the right hand-side equation (21) can evaluated by taking
21
and it becomes;
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
It has been established that
www.iiste.org
, hence;
Likewise taking
Therefore,
Substituting equations (21) through (24) back into equation (20), we obtain that;
At
And,
At
And,
The left hand-side of equations (26) and (28) governs the element’s strain distribution, and in terms of the finite
element method, it is the element property matrix.
Recalling from equations (13) and (14),
22
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
Also from equation (12)
Substituting equations (29) and (30) into the left hand-side terms of equations (26) and (28);
At
At
Combining equations (31) and (32) and expressing the out come in matrix form, thus;
We can now substitute the above result into equations (26) and (28) and expressing the result in matrix form to give
us the final version of element equations starting with element 1, thus;
Assigning values to equation (33), as
, creating our system topology,
23
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
TABLE 2: The System Topology for the Finite Element Segmentation Scheme from Fig (2)
Element
Node Numbers
Global
Local
1
1
2
1
2
2
2
1
3
2
3
3
4
1
2
4
4
1
5
2
5
5
1
6
2
6
6
1
7
2
And starting our solving with the second term of the right hand-side of equation (33), thus;
At
24
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
And at
Then solving the left hand-side term of equation (33);
Combining the above results and substituting into the element equation (33), gives the following;
And,
Combining equations (34) and (35) gives us element 1 stiffness matrix which forms the element matrix for other
elements, and since we have 6 elements and 7 nodes, we will have a system of 7 equations like the one below;
For the assembly of all elements equations, we form a Zero 7 x 7 initialized stiffness matrix, thus;
25
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
Assembly of the elements equations based on global coordinate points, hence Element 1 assembly;
After assembly of Element 2, we have;
After assembly of Element 3, we have;
After assembly of Element 4, we have;
26
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
After assembly of Element 5, we have;
After assembly of Element 6, we have;
Element 6 stiffness matrix can be re-written as below, i.e completed assembly at node 7;
The equation depicts the assembly matrix of the 6 elements at 7 nodes. Applying the initial boundary conditions,
where
were given as;
to obtain the values of
and computing into the assembly matrix, in order
The matrix below is obtained;
27
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
matrix was solved and the following result was
Therefore using Mat lab software and algorithm, the above
obtained;
Note: Above matrix could also be solved traditionally (numerically) using LU decomposition or Gaussian
elimination methods.
3. Concluding Remarks
At the assumption of a linear, one dimensional periodic boundary problem, the composite was discretized into six(6)
elements of 7 nodes. Through the application of the Galerkin’s method of weighted residual, the six elements’
stiffness matrices were assembled and the approximate numerical solution found.
Reference
[1]
Berner,
J.M.
(1993).
Finite
Element
Analysis
of
Damage
in
Fibrous
Composite
Using
a
Micromechanical mode. Masters’ Degree in Engineering Thesis, Naval Postgraduate School, Monterey, Califonia,
U.S.A.
[2]
Grandin, H. (1991). Fundamental of the Finite Element Method, Waveland Press.
[3] Finlayson, B.A., and Scriven, L.E. (1966). The Method of Weighted Residual – a Review. Applied Mechanics
Reviews, Vol. 19, No. 19, 735-748.
[4] Crandall, S.H. (1956). Engineering Analysis. McGraw Hill Publishng, AMR 12(1959), Rev. 1122.
[5] Ames, W.F. (1965). Nonlinear Partial Differential Equations in Engineering. Academic Press, AMR 19(1966),
28
Innovative Systems Design and Engineering
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.4, No.4, 2013
www.iiste.org
Rev. 4724.
[6] Collatz, L. (1960). The Numerical Treatment of Differential Equations. Springer Press, AMR 14(1961), Rev.
1738.
[7] Bensoussan, A., Lions, J.L., and Papanicolaon, G. (1978). Assmptototic Analysis for Periodic Structures. Studies
in Mathematics and its Applications, Vol. 5, North-Holland, Amsterdam.
[8] Sanchez-Palencia, E. (1980). Non-homogeneous media and vibration theory, No. 127 in Lecture Notes in Physics,
Springer-Verlag, Berlin.
[9] Galerkin, B.G., and Scerzhni, I.P. (2015). Series Occurring in Some Problems of Elastic Equilibrium of Rods and
Plates. Fed. Sci. Tech. Info., 19, 897-908.
[10] Kravchuk, M.F. (1952). Application of the Method of Moments to the Solution of Linear Differential and
Integral Equations. Kiev, Ukraine, UKSSR 169.
[11] Kantorovich, L.A. (1933). A Direct Method of Solving of Problem of the Minimum of a Double Integral. Int.
Akad. Nauk. SSSR 5, 647-652.
[12] Frazer, R.A., Jones, W.P., and Skan, S.V. (1937). Approximations to Functions and to the Solutions of
Differential Equations. Gt. Brit. Air Ministry Aero. Res. Comm. Tech. Rept. 1, 517-549.
[13] Poritzky, H. (1938). The Reduction of the Solution of Certain Partial Differential Equations to Ordinary
Differential Equations. Trans. Fifth Inter. Congr. Appl. Mech., Cambridge, Mass., pp 700-707.
[14] Repman, Yu. V. (1940). A Problem in the Mathematical Foundation of Galerkin’s Method for Solving Problems
on the Stability of Elastic Systems. Prikl. Mat. Mekh., 4 (2), pp 3-6.
[15] Bickley, W.G. (1941). Experiments in Approximating to the Solution of Partial Differential Equations. Phil.
Mag. , 7 (32), 50-56.
[16] Keldysh, M.V. (1942). On B.G. Galerkin’s Method for the Solution of Boundary-Value Problems. Izv. Akad.
Nauk. Ser. Mat., 6, 309-330.
[17] Yamada, H. (1947). A Method of Approximate Integration of the Laminar Boundary-Layer Equation. Rept. Res.
Inst. Fluid Engng., Kyushu Univ., 2, 29.
[18] Green, J.W. (1953). An Expansion Method for Parabolic Partial Differential Equations. Journal Res. Natl. bur.
Stds., 51, 127-132, U.S.A.
29
This academic article was published by The International Institute for Science,
Technology and Education (IISTE). The IISTE is a pioneer in the Open Access
Publishing service based in the U.S. and Europe. The aim of the institute is
Accelerating Global Knowledge Sharing.
More information about the publisher can be found in the IISTE’s homepage:
http://www.iiste.org
CALL FOR PAPERS
The IISTE is currently hosting more than 30 peer-reviewed academic journals and
collaborating with academic institutions around the world. There’s no deadline for
submission. Prospective authors of IISTE journals can find the submission
instruction on the following page: http://www.iiste.org/Journals/
The IISTE editorial team promises to the review and publish all the qualified
submissions in a fast manner. All the journals articles are available online to the
readers all over the world without financial, legal, or technical barriers other than
those inseparable from gaining access to the internet itself. Printed version of the
journals is also available upon request of readers and authors.
IISTE Knowledge Sharing Partners
EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open
Archives Harvester, Bielefeld Academic Search Engine, Elektronische
Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial
Library , NewJour, Google Scholar
View publication stats